Why $\Bbb Z$ = (integers) is an integral domain?

Question

Is there a proof for $\Bbb Z$ being an integral domain? Or is it an axiom that if $a*b=0$ then $a=0$ or $b=0$ where $a$, $b$ belong to $\Bbb Z$. Is it so obvious?

Answer 1

Just prove that if $a,b\ne0$, then $ab\ne0$. Which is quite obvious, because for positive $b$ you have $$a\cdot b=\underbrace{a+\ldots+a}_{b\text{ times}}\ne0$$ Because it's either $\geq a$ or $\leq a$ depending on the sign of $a$.

For $b<0$ just consider $(-a)(-b)$.

Answer 2

If you can use that $\mathbb Z \subset \mathbb Q$ then it is clear because every subring of a field is an integral domain.